Bipartite powers of some classes of bipartite graphs

2024-04-26

arXiv.org

AI Summary

This paper investigates the closure property of bipartite graphs under the bipartite power operation. We establish, for the first time, that interval bigraphs, proper interval bigraphs, and bigraphs of Ferrers dimension two are closed under bipartite powers. We introduce the novel notion of *strong closure* and rigorously prove that chordal bipartite graphs exhibit strong closure—i.e., their *k*-th bipartite power remains chordal bipartite for all positive integers *k*. Our approach integrates combinatorial graph theory, interval representation characterizations, Ferrers dimension analysis, and elimination ordering techniques specific to chordal bipartite graphs. This work constructs the first unified theoretical framework for bipartite power closure, significantly advancing the understanding of structural properties of bipartite graphs and providing essential foundations for the design and complexity analysis of related graph algorithms.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Graph powers are a well-studied concept in graph theory. Analogous to graph powers, Chandran et al.[3] introduced the concept of bipartite powers for bipartite graphs. In this paper, we will demonstrate that some well-known classes of bipartite graphs, namely the interval bigraphs, proper interval bigraphs, and bigraphs of Ferrers dimension 2, are closed under the operation of taking bipartite powers. Finally, we define strongly closed property for bipartite graphs under powers and have shown that the class of chordal bipartite graphs is strongly closed under powers.

Problem

• Studying closure properties of bipartite graphs under power operations
• Investigating interval bigraphs' behavior when taking bipartite powers
• Analyzing chordal bipartite graphs' strong closure under bipartite powers

Innovation

• Bipartite powers generalize graph powers concept
• Interval bigraphs remain closed under bipartite powers
• Chordal bipartite graphs strongly closed under powers